A conformal invariant for non-vanishing analytic functions and its applications

Abstract

The quantity $V_D(\varphi)=\sup_{z\in D}\rho_D(z)\inv |\varphi'(z)/\varphi(z)|$ will be considered for a non-vanishing analytic function $\varphi$ on a plane domain $D$ with hyperbolic metric $\rho_D(z)|dz|.$ We see that this quantity has various nice properties such as conformal invariance and monotoneity. As a special case, for a proper subdomain $\Omega$ of the punctured plane $\C^*=\C\setminus\{0\},$ we define the domain constant $W(\Omega)=V_\Omega(\id),$ which will be called the circular width of $\Omega$ about the origin, and we will see that $W(\Omega)$ dominates the value of $V_D(\varphi)$ if $\varphi(D)\subset\Omega.$ As applications, we provide boundedness and univalence criteria for those functions $f$ on the unit disk $\D$ for which $f'(\D)\subset\Omega.$ We also compute values of circular width for typical domains.

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